Cayleys Sextic
The curve, Cayley’s Sextic can be described by the Cartesian equation: 4(x^2 + y^2 – ax)^3 = 27a^2(x^2 + y^2)^2. It is the involute of a nephroiod curve because of its slight kidney shape and because they are parallel curves. This curve was first discovered by a mathematician by the name of Colin Maclaurin. Maclaurin who was born in February of 1698, became a student at Glasgow University in Scotland during his early teen years. It was here that he discovered his abilities in mathematics and began working towards a future in geometry and mathematics. In 1717 Maclaurin was given the job as the professor of mathematics at Marischal College in the University of Aberdeen. Later during his mathematical career, Maclaurin wrote Geometrica Organica, a book which displayed early ideas of what later becomes known as the curve, Cayley’s Sextic.
The actual man credited with the distinct discovery of Cayley’s Sextic is the man it is named after, Arthur Cayley. Cayley, who had a family of English ancestry, lived in St. Petersburg, Russia during his childhood where he attended his first years of schooling. In 1835 he began attending King’s College School in England because of his promise as a mathematician. After Cayley became a lawyer and studied math during his spare time, publishing papers in various mathematical journals. These journals were later looked at by Archibald and in a paper published in 1900 in Strasbourg he gave Cayley the honor of having the curve named after him.
Cayley’s Sextic
The polar form of the equation for the curve, Cayley’s Sextic, is shown as:
r = 4a cos^3 (q/3). For the specific equation for the graph, the polar form is the equation of greatest ease of use. Use 1 in place of “a” and switch the calculator to polar form. The best viewing window for this graph is q min= -360; q max= 360; q step= 10; x-min= -5; x-max= 5; x scale= 1; y-min= -5; y-max= 5; y scale= 1. This window and equation will give an excellent picture of the curve, Cayley’s Sextic.
When “a” is increased in the equation for the curve, the entire curve increases in size, giving it a larger area. The value for “x” is greatly increased on the right side positive y-axis, while the value for “x” on the left side negative y-axis becomes gradually more negative at a much lower rate then that of the right side positive y-axis.
Overall, as the concentration of the substrate increases, the enzyme activity increases up to a 70% of solution, where the enzyme activity starts to level off. The curve is polynomial because of the fact that the enzyme activity exponentially increases as the concentration of substrate increase; additional evidence for this is the fact that the gradient graph is constantly changing. The polynomial curve is shown because until 70% (the saturation point); this is because there are more casein substrate molecules that can successfully collide with the renin enzyme molecule, therefore increasing the rate of reaction.
...does increase, proving the fact that the parabola not only become more definite in shape as the distance increases, but so does the trajectory and height.
O'Connor, J. J., and E. F. Robinson. "Hopper Biography." MacTutor History of Mathematics. University of St Andrews, July 1999. Web. 29 Sept. 2011. .
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Below the graph consist of the intersections of the line ‘y = x’ , ‘y = 2x’ with the curve y = x4
On January 27, 1832 Charles Lutwidge Dodgson was born in Daresbury, Cheshire Country, England. In 1943 his family moved to the croft Rectory in Richmondshire, North Yorkshire, while he was enrolled at the Richmond public school. Three years later at the age of fourteen in the year of 1846, he went on to the Rugby school in Warwickshire. He spent three years at the school in Warwickshire and left in the year of 1849. Later he went to Oxford in 1851 and earned a B.A. with first class honors in mathematics and second class in classics in 1854. Several years later in 1857 he graduated with an M.A. finishing his studies at oxford. The year 1856 was advent of the use “Lewis Carroll” an Anglicized pseudonym, which he took to publish all his literary works. Mirroring his father’s career path, he obtained the position of Mathematical Lecturer at Oxford which he maintained from 1856 to 1881. Year 1861 he received holy orders, becoming a deacon at the Christ Church Cathedral, however he was unable to be ordained a priest due to his lack of interest in ministration. In 1865 he published the novel Alice’s Adventures in Wonderland, his most renowned literary pieces that is still talked about to this day. Four years later he published Phantasmagoria, a ten year collection of poems, and seven years after that was The Hunting of the Snark. All work associated with his knowledge of mathematics, such as Two Books of Euclid, Elementary Treatise on...
the root to the function, like if it is a parabola with its vertex is placed
In 1665, the Binomial Theorem was born by the highly appraised Isaac Newton, who at the time was just a graduate from Cambridge University. He came up with the proof and extensions of the Binomial Theorem, which he included it into what he called “method of fluxions”. However, Newton was not the first one to formulate the expression (a + b)n, in Euclid II, 4, the first traces of the Binomial Theorem is found. “If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle of the segments” (Euclid II, 4), thus in algebraic terms if taken into account that the segments are a and b:
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Fermat was born in 1601 in Beaumont-de-Lomagne, France and initially studied mathematics in Bordeaux with some of the disciples of Viete, a French algebraist (Katz 2009). He went on to earn a law degree and become a successful counselor. Mathematics was merely a hobby to him, so he never published because he did not want to thoroughly explain his discoveries in detail. He died in 1665 and his son later published his manuscripts and correspondence. Fermat adapted Viète’s algebra to the study of geometric loci and used letters to represent variable distances. He discovered that the study of loci, or sets of points with certain characteristics, could be made easier by applying algebra to geometry through a coordinate system (Katz 2009). Basically any relation between ...
Robert Hooke was born on July 18th, 1635 in Freshwater, Isle of Wight, England. His father, John Hooke, was a clergyman. As a child Hooke became ill of smallpox, of which he survived from, only to be disfigured and scarred. Throughout his childhood, Robert never really received much of any regular schooling due to his sickness and weakness. On the other hand he had an amazing natural curiosity, which led to the development of his mind through self-learning. When Robert was merely thirteen years old his father committed suicide by hanging himself. All that was left behind for Robert was 40 pounds. After his father died, Hooke was sent to London as an orphan, where he studied under Peter Lely, an artist of the time. He soon realized that he should spend his inheritance attending Westminster School, where he lodged with Dr. Richard Rusby. Robert had a large interest in mechanical objects and was encouraged greatly by Dr. Busby. Within the first week of being with Dr. Busby, Hooke was able to work through many books of Euclid's geometry. He was soon allowed unsupervised access to Dr. Busby's library. When Robert was eighteen he moved on and attended Oxford, where he soon after obtained his masters degree. Once he secured the sponsorship and guidance of John Wilkins, the warden of Wadham College, he was well on his way to become one of the greatest inventors, microscopists, physicists, surveyors, astronomers, biologists, artists.
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...
Abstractions from nature are one the important element in mathematics. Mathematics is a universal subject that has connections to many different areas including nature. [IMAGE] [IMAGE] Bibliography: 1. http://users.powernet.co.uk/bearsoft/Maths.html 2. http://weblife.bangor.ac.uk/cyfrif/eng/resources/spirals.htm 3.